Theorem nfrexya 2477

Description: Not-free for restricted existential quantification where 𝑦 and 𝐴 are distinct. See nfrexxy 2475 for a version with 𝑥 and 𝑦 distinct instead. (Contributed by Jim Kingdon, 3-Jun-2018.)

Hypotheses

Ref	Expression
nfralya.1	⊢ Ⅎ𝑥𝐴
nfralya.2	⊢ Ⅎ𝑥𝜑

Assertion

Ref	Expression
nfrexya	⊢ Ⅎ𝑥∃𝑦 ∈ 𝐴 𝜑

Distinct variable group:   𝑦,𝐴

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥)

Proof of Theorem nfrexya

Step	Hyp	Ref	Expression
1		nftru 1443	. . 3 ⊢ Ⅎ𝑦⊤
2		nfralya.1	. . . 4 ⊢ Ⅎ𝑥𝐴
3	2	a1i 9	. . 3 ⊢ (⊤ → Ⅎ𝑥𝐴)
4		nfralya.2	. . . 4 ⊢ Ⅎ𝑥𝜑
5	4	a1i 9	. . 3 ⊢ (⊤ → Ⅎ𝑥𝜑)
6	1, 3, 5	nfrexdya 2473	. 2 ⊢ (⊤ → Ⅎ𝑥∃𝑦 ∈ 𝐴 𝜑)
7	6	mptru 1341	1 ⊢ Ⅎ𝑥∃𝑦 ∈ 𝐴 𝜑

Syntax hints:  ⊤wtru 1333  Ⅎwnf 1437  Ⅎwnfc 2269  ∃wrex 2418

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-cleq 2133  df-clel 2136  df-nfc 2271  df-rex 2423

This theorem is referenced by:  nfiunya  3849  nffrec  6301  nfsup  6887  caucvgsrlemgt1  7627  nfsum1  11157  zsupcllemstep  11674  bezout  11735